Fall 2011, Math 101

November and December, 2011

Be sure to check back often, because assignments may change!

 All section and page numbers refer to sections from Calculus: Early Transcendental Functions, 3rd Edition, by Smith and Minton.

I'll use Maple syntax for some of the mathematical notation on this page. (Paying attention to how I type various expressions is a good way to absorb Maple notation). I will not use it when I think it will make the questions too difficult to read.

Due Wednesday 11/2 at 8am

Section 3.7: Optimization

To read: Thru Example 3.7.3. Really read through these examples carefully, paying attention to why the authors are doing each step. In Example 3.7.3, pay particular attention to the transitoin from the distance function d(x) to the function to be optimized, f(x)=[d(x)]2 -- this is a useful technique

In Example 3.7.1 and 3.7.2, the authors caution us not to simplify the function to be maximized out of habit ; although they omit this caution in Example 3.7.3 they again refrain from expanding the function to be maximized.
1. In Example 3.7.1, would expanding A(x) have made any of the steps that followed simpler? Would it have made any of the steps harder?
2. In Example 3.7.2, would expanding V(x) have made any of the steps easier? Harder?

Reminder:

• Bring questions on PS 9.
• You have a reading assignment due Thurday also. I decided to split Section 3.7 up, as I thought it might be conceptually a bit much to read all at once.
• If you are still working on the Differentiation Exam, the next deadline is 11/28, to receive 75%.

Due Thursday 11/3 at 8am

Section 3.7: Optimization (continued)

To read: Finish the section. Pay close attention to Remark 3.7.1.

1. In Example 3.7.5,
1. what physically does the quantity 2πr2 represent? How about 2πrh?
2. The derivative A'(r) is not defined at r=0. Why is 0 not a critical number for A(r)?
3. the result the authors arrived at for the values that minimize the amount of material used in a can do not reflect the reality of the design of most 12 oz cans. Did you notice any unrealistic assumptions that were made while working through the problem that may explain this difference?
2. In Example 3.7.6, why are the only critical numbers those values of x where C '(x)=0?
3. For each of the following, answer True or False, and give a brief explanation or example for your choice
1. If there is only one critical point, it must be the minimizer or maximizer that you are seeking.
2. If there is only one critical point, it must be a local extremum.
3. If there is only one local extremum, it must be an absolute extremum as well.

Due Friday 11/4 at 8am

Handout -- Ostebee and Zorn, Section 4.7: Taylor Polynomials (available on OnCourse page)

1. Why might finding the Taylor polynomial of a function be useful?
2. In your own words, briefly explain the idea (not just the steps) of building a Taylor polynomial for a function f(x).

Reminder:

• Begin working on WW 10 and PS 10 (another group problem set). Once again, switch partners, be primary author if you weren't last time, and don't split up the problems.

Due Monday 11/7 at 8am

3.7: Optimization

To read: Re-read the examples, really paying attention to how they set the problems up.

• For those who are choosing to redo problems on Exam 2 that they missed, for extra credit - Remember it's due Friday 11/11 at 2pm.

Reminder:

Due Wednesday 11/9 at 8am

Handout -- Ostebee and Zorn, Section 4.7: Taylor Polynomials (continued) (available on OnCourse page)

Reminder:

• Bring questions on PS 10.
• Remember to put a star next to the primary author's name.
• For those who are choosing to redo problems on Exam 2 that they missed, for extra credit - Remember it's due Friday 11/11 at 2pm.

Due Friday 11/11 at 8am

Project 2 (continued)

To read: You should have read through the entire letter before lab on Thursday. Before class on Friday, re-read the letter, and review the work you did in lab on Thursday.

Reminder:

• Begin WW 11 and PS 11.
• For those who are choosing to redo problems on Exam 2 that they missed, for extra credit - Remember it's due Friday 11/11 at 2pm.
• Remember: Because this is your second project (and because I thought the differentiation exam might interfere with your first project), you have considerably less time to work on this project than on the first one, so use your time wisely.

Due Monday 11/14 at 8am

Handout --Ostebee and Zorn, Section 5.1: Areas and Integrals (on OnCourse)

To read: All. Be sure to understand the definition of the integral, Example 2, and the section "Properties of the Integral" beginning on page 306.

1. What does the integral of a function f from x=a to x=b measure?
2. Is the integral of f(x)=5x from x=-1 to x=3 positive or negative?

Reminder:

• Aim to be done with the calculations for the project by Tuesday afternoon.

Due Wednesday 11/16 at 8am

Handout-- Ostebee and Zorn, Section 5.2: The Area Function (on OnCourse)

To read: All. Make sure you understand the definition of the area function and Examples 2, 3, and 4.

1. Let f be any function. What does the area function Af(x) measure?
2. Let f(t)=t and let a=0. What is Af(1)?

Reminder:

• Bring remaining questions on PS 11 to class.
• Continue meeting with your group for Project 2; get started writing your response letter.
• Exam 3 will be on Thursday 12/1, the Thursday after Thanksgiving Break.

Due Friday 11/18 at 8am

Handout -- Ostebee and Zorn, Section 5.3: The Fundamental Theorem of Calculus (on OnCourse)

To read: All, but you can skip the proof of the FTC if you'd like: we'll look at a different approach in class.

1. Find the area between the x-axis and the graph of f(x)=x^3+4 from x=0 to x=3.
2. Does every continuous function have an antiderivative? Why or why not?
3. If f(x)=3*x-5 and a=2, where is Af increasing? Decreasing? Why?

Reminder:

• Begin working on WW 12. As usual, the Study Guide for Exam 3 will assume you have worked through all the problems on it.
• Aim to have a solid rough draft of your response letter for Project 2 done by Friday afternoon.
• If you are still working on the Differentiation Exam, the deadline to receive 75% on it is the Monday after Thanksgiving Break, Monday 11/28 at 5pm.

Due Monday 11/21 at 8am

Handout --Ostebee and Zorn, Section 5.3: The Fundamental Theorem of Calculus

Reminder:

• Project 2 is due at 5pm Monday.
• If you are still working on the Differentiation Exam, the deadline to receive 75% on it is next Monday, 11/28, at 5pm.

Due Wednesday 11/23 at 8am

Thanksgiving Break

No Reading Questions Today, Of Course

Due Friday 11/25 at 8am

Thanksgiving Break

Due Monday 11/28 at 8am

Section 4.1: Antiderivatives
Section 4.6: Integration by Substitution

To read: In Section 4.1, the only new material is on page 346 (the definition of the indefinite integral). However, if you're feeling shaky on antiderivatives, it would be wise to read the whole section. In Section 4.6, read all except for Example 4.6.8.

1. What do ex2 from Example 4.6.1, (x3+5)100 from Example 4.6.2, cos(x2) from Example 4.6.3, (3sin x +4)5 from Example 4.6.4, and sin(√x) from Example 4.6.5 all have in common?
2. Looking again at these same functions and examples -- what did the choice of u have in common in each?
3. When calculating antiderivatives or indefinite integrals, how can you always check whether you result is correct (without using a computer or a calculator?)

Reminders:

• The deadline to receive 75% on the differentiation exam is Monday at 5pm.
• As usual, put plenty of time and effort into studying for Exam 3; I urge you once again to spread your studying out over more than just the last two days.

Due Wednesday 11/30 at 8am

Bring Questions for Exam 3

To read: As before, I ask you to review the Honor Code, Wheaton's description of plagiarism, and the portion in the course policies that applies to the Honor Code as a reminder of how important a role the Honor Code plays at Wheaton.

Reminders:

• Visit my office hours and visit the Kollett Center while clearing up questions before the exam!
• Get questions on WW 12 out of the way before class. WW12 should be done before class so that you can focus on reviewing.
• As usual, you may have a "cheat sheet", consisting of handwritten notes on one side of an 8 1/2 x 11 (or smaller) piece of paper for the exam.
• As before, you may begin taking the exam at 12:30pm Thursday.

Due Friday 12/2 at 8am

Section 4.6: Substitution

Reminders:

• Begin WW 13.
• If you are still working on the Differentiation Exam, the last day to receive any credit (50%) is the last day of classes (Friday 12/9), at 3pm.

Due Monday 12/5 at 8am

Handout -- Section 5.6: Approximating Sums: The Integral as a Limit (on OnCourse)

To read: All. Be sure to understand the definitionof a Riemann Sum and Example 3.

1. Explain, in your own words, the idea of using Riemann Sums to approximate integrals.
2. If f(x) is decreasing on [a,b], will Ln underestimate or overestimate the integral of f from a to b? How about Rn?

Due Wednesday 12/7 at 8am

Section 4.2: Sums and Sigma Notation

To read: You may skip the beginning of Section 4.2. Begin reading in the middle of page 355, with the paragraph that begins with the sentence "We begin by introducing some notation." Read through Example 4.2.3, and also just read the statement of Theorem 2.2.

1. Write 2-1+3-1+4-1+5-1+6-1+7-1+8-1+9-1+10-1+11-1+12-1 in summation notation.
2. Write out the terms (do not evaluate) from expanding
3. What do you think the purpose of summation (sigma) notation is?
Reminders:
• I will answer questions on WW 13 on Thursday (or possibly Friday) rather than on Wednesday as is usual.

Due Friday 12/10 at 8am

The Big Picture

To read: Once again, I remind you of the importance of the Honor Code, and ask you to re-read the it, Wheaton's description of plagiarism, and the portion in the course policies that applies to the Honor Code. Pay particular attention to how all of this applies to exam situations.

Reminders:

• The deadline to receive 50% on the Diff Exam is Friday at 3pm.
• If I did not deal with questions on WW 13 on Thursday, then bring them to class Friday.

Janice Sklensky
Wheaton College
Department of Mathematics and Computer Science
Science Center, Room 327
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278
jsklensk@wheatonma.edu

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